Abstract: Distribution, redistribution, recycling and repacking have become an important issue in logistic planning duringthe last decades. While keeping operational cost as low as possible still the main goal for logistic planners, other aspectssuch as recycling are getting more attention from industry. In this article the well known Ant Colony System (ACS), a bioinspiredalgorithm, is implemented to solve a problem arising in Reverse Logistic namely Vehicle Routing Problem withSimultaneous Delivery and Pickup (VRPSDP). To solve this problem we need to find the optimal set of paths that meet, atthe same time, customer delivery and pickup demands. In order to solve this problem, our ACS implementation makes useof a strategy that mimics the effect of the pheromone in the natural Ants behaviour. To do that, each vehicle is viewed asan individual agent (ant) and consequently its behaviour is driven by pheromone strategy, i.e. it tends to choose the routefor which the pheromone level is higher. Results show that our ACS implementation provides good quality solutionswithin an acceptable time. Furthermore, obtained solutions are quite competitive when compared to other stochastictechniques previously studied in literature.

During the last few decades an increasing number of environmental issues have led to changes in the normative that regulates the logistic industry. In particular, new requirements from customers such as recycling and proper items disposal cause that logistic industry should not only look at the delivery process but also to the pick-up one. Because of that, well-known Vehicle Routing Problem (VRP) is not an adequate model to the integrated problem mentioned before. Therefore, several new models have been introduced in order to address this new scenario. In this article we consider one of those models namely VRP with simultaneous delivery and pick-up (VRPSDP). VRPSDP problem has been firstly introduced and modelled in [1]. In [1] authors studied the VRPSDP based on a real case from book distribution industry. They tackled the book logistic activity from one central library to a set of local libraries and vice-versa. They considered a fixed number of vehicles and a limited vehicle capacity. Different mathematical models for the VRPSDP have been proposed in [2], [3] and [4]. Particularly, authors in [2] modelled the VRPSDP as part of the reverse logistics process. In this paper we attempt to find high quality solutions to this optimisation problem using the well-known Ant Colony Systems (ACS) heuristic.

Ant Colony Systems has demonstrated to be very effective for routing problems [9]. In general it is able to find near-optimal routes for many problems such as VRP [14, 15, 20] and travelling salesman problem [10, 12, 16], among others. One advantage of ACS over other heuristics is its rapid convergence, which means high quality solutions within an acceptable time. To the best of our knowledge, ACS algorithm has only been used to solve the VRPSDP problem in [15, 21]. Although similar strategies are considered, our algorithm implements quite different steps and rules which make our approach substantially different from those previously presented approaches. This article is organised as follows: Section 2 presents a literature review and introduces the mathematical model for the VRPSDP that is considered in this study. Section 3 reviews ACS algorithms and describes in detail the proposed ACS algorithm for the VRPSPD. Section 4 presents the benchmark used in this study. Obtained results are analysed at the end of this section. Finally, some conclusions are outlined in Section 5.